TheInfoList

The square root of 2 (approximately 1.4142) is a positive
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
that, when multiplied by itself, equals the . It may be written in mathematics as $\sqrt$ or $2^$, and is an
algebraic number An algebraic number is any complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
. Technically, it should be called the principal
square root In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

. It was probably the first number known to be
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small
denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
. Sequence in the
On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. He transferred the intellectual property Intellectual property (I ...
consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: :

# History

The
Babylonia Babylonia () was an and based in central-southern which was part of Ancient Persia (present-day and ). A small -ruled state emerged in 1894 BCE, which contained the minor administrative town of . It was merely a small provincial town dur ...
n clay tablet
YBC 7289 YBC 7289 is a Babylonia Babylonia () was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia Mesopotamia ( ar, بِلَاد ٱلرَّافِدَيْن '; grc, Μεσοποταμία; Syriac l ...
(c. 1800–1600 BC) gives an approximation of in four
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
figures, , which is accurate to about six
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
digits, and is the closest possible three-place sexagesimal representation of : :$1 + \frac + \frac + \frac = \frac = 1.41421\overline.$ Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC), as follows: ''Increase the length f the sideby its third and this third by its own fourth less the thirty-fourth part of that fourth.'' That is, :$1 + \frac + \frac - \frac = \frac = 1.41421\overline.$ This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of
Pell number Pell is a surname shared by several notable people, listed below * Axel Rudi Pell (born 1960), German heavy metal guitar player and member of Steeler and founder of his own eponymous band * Charles Pell (1874–1936), American college football c ...
s, which can be derived from the
continued fraction In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
expansion of . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.
Pythagoreans Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...
discovered that the diagonal of a
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...

is incommensurable with its side, or in modern language, that the square root of two is
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
. Little is known with certainty about the time or circumstances of this discovery, but the name of
Hippasus Hippasus of Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Taranto, Tarentum, between the river Bradanus and the Casuentus (modern Basento). It wa ...
of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant, for example by .

## Ancient Roman architecture

In
ancient Roman architecture , Spain, a masterpiece of ancient bridge building , a Roman theatre in Athens, Greece Ancient Roman architecture adopted the external language of classical Ancient Greek Architecture, Greek architecture for the purposes of the ancient Ro ...
,
Vitruvius Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...

describes the use of the square root of 2 progression or ''ad quadratum'' technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to
Plato Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thoug ...

. The system was employed to build pavements by creating a square
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

to the corners of the original square at 45 degrees of it. The proportion was also used to design
atriaAtria may refer to: *Atrium (heart) The atrium (Latin ātrium, “entry hall”) is the upper chamber through which blood enters the Ventricle (heart), ventricles of the heart. There are two atria in the human heart – the left atrium receives bloo ...
by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.

# Decimal value

## Computation algorithms

There are a number of
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s for approximating as a ratio of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the
Babylonian method Methods of computing square roots are numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... ...

for computing square roots, which is one of many
methods of computing square roots Methods of computing square roots are numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... ...
. It goes as follows: First, pick a guess, ; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the science, scientific study of language. It e ...

computation: :$a_ = \frac=\frac+\frac.$ The more iterations through the algorithm (that is, the more computations performed and the greater ""), the better the approximation. Each iteration roughly doubles the number of correct digits. Starting with , the results of the algorithm are as follows: * 1 () * = 1.5 () * = 1.416... () * = 1.414215... () * = 1.4142135623746... ()

## Rational approximations

A simple rational approximation (≈ 1.4142857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than (approx. ). The next two better rational approximations are (≈ 1.4141414...) with a marginally smaller error (approx. ), and (≈ 1.4142012) with an error of approx . The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with () is too large by about ; its square is ≈ .

## Records in computation

In 1997 the value of was calculated to 137,438,953,444 decimal places by
Yasumasa Kanada was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of pi, . He set the record 11 of the past 21 times. Kanada was a professor in the Department of Information Science at ...
's team. In February 2006 the record for the calculation of was eclipsed with the use of a home computer. Shigeru Kondo calculated 1
trillion A trillion is a number with two distinct definitions: * 1,000,000,000,000, i.e. one million million, or (ten to the twelfth power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted ...
decimal places in 2010. Among
mathematical constant A mathematical constant is a key whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an ), or by mathematicians' names to facilitate using it across multiple s. Constants arise in many areas of , with constan ...
s with computationally challenging decimal expansions, only

has been calculated more precisely. Such computations aim to check empirically whether such numbers are normal. This is a table of recent records in calculating the digits of .

# Proofs of irrationality

A short
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
of the irrationality of can be obtained from the
rational root theorem In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
, that is, if is a monic
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

with integer
coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, then any
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large group ...
of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a perfect square), must therefore be irrational. This proof can be generalized to show that any square root of any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see
Quadratic irrational numberIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
or
Infinite descentIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

## Proof by infinite descent

One proof of the number's irrationality is the following
proof by infinite descentIn mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for ...
. It is also a
proof by contradiction In logic and mathematics, proof by contradiction is a form of Mathematical proof, proof that establishes the Truth#Formal theories, truth or the Validity (logic), validity of a proposition, by showing that assuming the proposition to be false leads ...
, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true. # Assume that is a rational number, meaning that there exists a pair of integers whose ratio is exactly . # If the two integers have a
factor FACTOR (the Foundation to Assist Canadian Talent on Records) is a private non-profit organization "dedicated to providing assistance toward the growth and development of the Music of Canada, Canadian music industry". FACTOR was founded in 1982 by r ...

, it can be eliminated using the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
. # Then can be written as an
irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquial ...
such that and are
coprime integers In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
(having no common factor) which additionally means that at least one of or must be odd. # It follows that and .   (  )   ( are integers) # Therefore, is even because it is equal to . ( is necessarily even because it is 2 times another whole number.) # It follows that must be even (as squares of odd integers are never even). # Because is even, there exists an integer that fulfills . # Substituting from step 7 for in the second equation of step 4: is equivalent to , which is equivalent to . # Because is divisible by two and therefore even, and because , it follows that is also even which means that is even. # By steps 5 and 8 and are both even, which contradicts that is irreducible as stated in step 3. ::'' Q.E.D.'' Because there is a contradiction, the assumption (1) that is a rational number must be false. This means that is not a rational number. That is, is irrational. This proof was hinted at by
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

, in his '' Analytica Priora'', §I.23. It appeared first as a full proof in
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's '' Elements'', as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an
interpolation In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...
and not attributable to Euclid.

## Proof by unique factorization

As with the proof by infinite descent, we obtain $a^2 = 2b^2$. Being the same quantity, each side has the same prime factorization by the
fundamental theorem of arithmetic In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
, and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.

## Geometric proof

A simple proof is attributed by
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when con ...
to
Stanley Tennenbaum Stanley Tennenbaum (April 11, 1927 – May 4, 2005) was an American mathematician who contributed to the field of logic. In 1959, he published Tennenbaum's theorem, which states that no countable set, countable Non-standard model of arithmetic, non ...
when the latter was a student in the early 1950s and whose most recent appearance is in an article by Noson Yanofsky in the May–June 2016 issue of ''
American Scientist __NOTOC__ ''American Scientist'' (informally abbreviated ''AmSci'') is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society. In the beginning of 2000s the headquarters was in New H ...
''. Given two squares with integer sides respectively ''a'' and ''b'', one of which has twice the
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle () must equal the sum of the two uncovered squares (). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1. Another geometric
reductio ad absurdum In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents sta ...
argument showing that is irrational appeared in 2000 in the
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located i ...
. It is also an example of proof by infinite descent. It makes use of classic
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandr ...
construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the same algebraic proof as in the previous paragraph, viewed geometrically in another way. Let be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

, . Suppose and are integers. Let be a
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

given in its
lowest terms An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English ...
. Draw the arcs and with centre . Join . It follows that , and the and coincide. Therefore, the
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

s and are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
by SAS. Because is a right angle and is half a right angle, is also a right isosceles triangle. Hence implies . By symmetry, , and is also a right isosceles triangle. It also follows that . Hence, there is an even smaller right isosceles triangle, with hypotenuse length and legs . These values are integers even smaller than and and in the same ratio, contradicting the hypothesis that is in lowest terms. Therefore, and cannot be both integers, hence is irrational.

## Constructive proof

In a constructive approach, one distinguishes between on the one hand not being rational, and on the other hand being irrational (i.e., being quantifiably apart from every rational), the latter being a stronger property. Given positive integers and , because the (i.e., highest power of 2 dividing a number) of is odd, while the valuation of is even, they must be distinct integers; thus . Then :$\left, \sqrt2 - \frac\ = \frac \ge \frac \ge \frac,$ the latter
inequality Inequality may refer to: Economics * Attention inequality Attention inequality is a term used to target the inequality of distribution of attention across users on social networks, people in general, and for scientific papers. Yun Family Foundat ...
being true because it is assumed that (otherwise the quantitative apartness can be trivially established). This gives a lower bound of for the difference , yielding a direct proof of irrationality not relying on the
law of excluded middle In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
; see
Errett Bishop Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America ...
(1985, p. 18). This proof constructively exhibits a discrepancy between and any rational.

## Proof by Diophantine equations

* ''Lemma'': For the
Diophantine equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$x^2+y^2=z^2$ in its primitive (simplest) form, integer solutions exist
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
either $x$ or $y$ is odd, but never when both $x$ and $y$ are odd. ''Proof'': For the given equation, there are only six possible combinations of oddness and evenness for whole-number values of $x$ and $y$ that produce a whole-number value for $z$. A simple enumeration of all six possibilities shows why four of these six are impossible. Of the two remaining possibilities, one can be proven to not contain any solutions using
modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
, leaving the sole remaining possibility as the only one to contain solutions, if any. The fifth possibility (both $x$ and $y$ odd and $z$ even) can be shown to contain no solutions as follows. Since $z$ is even, $z^2$ must be divisible by $4$, hence :$x^2+y^2 \equiv 0 \mod4$ The square of any odd number is always
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
to 1 modulo 4. The square of any even number is always congruent to 0 modulo 4. Since both $x$ and $y$ are odd and $z$ is even: :$1+1 \equiv 0 \mod 4$ :$2 \equiv 0 \mod 4$ which is impossible. Therefore, the fifth possibility is also ruled out, leaving the sixth to be the only possible combination to contain solutions, if any. An extension of this lemma is the result that two identical whole-number squares can never be added to produce another whole-number square, even when the equation is not in its simplest form. * ''Theorem:'' $\sqrt2$ is irrational. ''Proof'': Suppose to the contrary $\sqrt2$ is rational. Therefore, :$\sqrt2 =$ :where $a,b \in \mathbb$ and $b \neq 0$ :Squaring both sides, :$2 =$ :$2b^2 = a^2$ :$b^2+b^2 = a^2$ But the lemma proves that the sum of two identical whole-number squares cannot produce another whole-number square. Therefore, the assumption that $\sqrt2$ is rational is contradicted. $\sqrt2$ is irrational. '' Q. E. D.''

# Multiplicative inverse

The
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

(reciprocal) of the square root of two (i.e., the square root of ) is a widely used
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
. :$\frac1 = \frac = \sin 45^\circ = \cos 45^\circ =$ ...   One-half of , also the reciprocal of , is a common quantity in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

because the
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
that makes a 45°
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

with the axes in a plane has the coordinates :$\left\left(\frac, \frac\right\right)\!.$ This number satisfies :$\tfrac\sqrt = \sqrt = \frac = \cos 45^ = \sin 45^.$

# Properties

One interesting property of is :$\!\ = \sqrt + 1$ since :$\left\left(\sqrt+1\right\right)\!\left\left(\sqrt-1\right\right) = 2-1 = 1.$ This is related to the property of
silver ratio In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
s. can also be expressed in terms of copies of the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...
using only the
square root In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

and
arithmetic operations Arithmetic (from the Greek ἀριθμός ''arithmos'', 'number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...
, if the square root symbol is interpreted suitably for the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s and : :$\frac\text\frac$ is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for , and for , the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of as will be called (if this limit exists) . Then is the only number for which . Or symbolically: :$\sqrt^ = 2.$ appears in for : : $2^m\sqrt \to \pi\textm \to \infty$ for square roots and only one minus sign. Similar in appearance but with a finite number of terms, appears in various trigonometric constants: :$\begin \sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt \\$\sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt \\\sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt \\\sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt \\\sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt &\quad \sin\frac &= \tfrac12\sqrt \end It is not known whether is a
normal numberNormal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Normal ...
, which is a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.

# Representations

## Series and product

The identity , along with the infinite product representations for the
sine and cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

, leads to products such as :$\frac = \prod_^\infty \left\left(1-\frac\right\right) = \left\left(1-\frac\right\right)\!\left\left(1-\frac\right\right)\!\left\left(1-\frac\right\right) \cdots$ and :$\sqrt = \prod_^\infty\frac = \left\left(\frac\right\right)\!\left\left(\frac\right\right)\!\left\left(\frac\right\right)\!\left\left(\frac\right\right) \cdots$ or equivalently, :$\sqrt = \prod_^\infty\left\left(1+\frac\right\right)\left\left(1-\frac\right\right) = \left\left(1+\frac\right\right)\!\left\left(1-\frac\right\right)\!\left\left(1+\frac\right\right)\!\left\left(1-\frac\right\right) \cdots.$ The number can also be expressed by taking the
Taylor series In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
of a
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. For example, the series for gives :$\frac = \sum_^\infty \frac.$ The Taylor series of with and using the
double factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
gives :$\sqrt = \sum_^\infty \left(-1\right)^ \frac = 1 + \frac - \frac + \frac - \frac + \cdots = 1 + \frac - \frac + \frac - \frac + \frac + \cdots.$ The
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
of this series can be accelerated with an
Euler transformIn combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the ...
, producing :$\sqrt = \sum_^\infty \frac = \frac +\frac + \frac + \frac + \frac + \frac + \cdots.$ It is not known whether can be represented with a BBP-type formula. BBP-type formulas are known for and , however. The number can be represented by an infinite series of
Egyptian fractions An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, ...
, with denominators defined by 2''n'' th terms of a
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathem ...

-like
recurrence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''a''(''n'') = 34''a''(''n''−1) − ''a''(''n''−2), ''a''(0) = 0, ''a''(1) = 6. :$\sqrt=\frac-\frac\sum_^\infty \frac=\frac-\frac\left\left(\frac+\frac+\frac+\dots \right\right)$

## Continued fraction

The square root of two has the following
continued fraction In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
representation: :$\!\ \sqrt = 1 + \cfrac.$ The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the
Pell number Pell is a surname shared by several notable people, listed below * Axel Rudi Pell (born 1960), German heavy metal guitar player and member of Steeler and founder of his own eponymous band * Charles Pell (1874–1936), American college football c ...
s (known as side and diameter numbers to the ancient Greeks because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: . The convergent differs from by almost exactly and then the next convergent is .

## Nested square

The following nested square expressions converge to : :$\begin \sqrt &=\tfrac - 2 \left\left( \tfrac- \left\left( \tfrac-\left\left( \tfrac- \left\left( \tfrac- \cdots \right\right)^2 \right\right)^2 \right\right)^2 \right\right)^2\\ &=\tfrac - 4 \left\left( \tfrac+ \left\left( \tfrac+\left\left( \tfrac+ \left\left( \tfrac+ \cdots \right\right)^2 \right\right)^2 \right\right)^2 \right\right)^2. \end$

# Applications

## Paper size

In 1786, German physics professor
Georg Christoph Lichtenberg Georg Christoph Lichtenberg (1 July 1742 – 24 February 1799) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area ...

found that any sheet of paper whose long edge is times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standardised
paper size Paper size Technical standard, standards govern the size of sheets of paper used as writing paper, stationery, cards, and for some printed documents. The ISO 216 standard, which includes the commonly used A4 size, is the international standa ...
s at the beginning of the 20th century, they used Lichtenberg's ratio to create the "A" series of paper sizes. Today, the (approximate)
aspect ratio The aspect ratio of a geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμό ...

of paper sizes under
ISO 216 ISO 216 is an International Organization for Standardization, international standard for paper sizes, used around the world except in North America and parts of Latin America. The standard defines the "A", "B" and "C" series of paper sizes, incl ...
(A4, A0, etc.) is 1:. Proof:
Let $S =$ shorter length and $L =$ longer length of the sides of a sheet of paper, with
:$R = \frac = \sqrt$ as required by ISO 216. Let $R\text{'} = \frac$ be the analogous ratio of the halved sheet, then
:$R\text{'} = \frac = \frac = \frac = \frac = \sqrt = R$.

## Physical sciences

There are some interesting properties involving the square root of 2 in the
physical sciences Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". ...
: * The square root of two is the
frequency ratio 9:8 major tone In Western culture, Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a interval (music), musical interval encompassing two adjacent sta ...
of a
tritone In music theory Music theory is the study of the practices and possibilities of music Music is the art of arranging sounds in time through the elements of melody, harmony, rhythm, and timbre. It is one of the universal cultural aspects ...

interval in twelve-tone
equal temperament An equal temperament is a or , which approximates by dividing an (or other interval) into equal steps. This means the ratio of the of any adjacent pair of notes is the same, which gives an equal perceived step size as is perceived roughly a ...
music. * The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of ''areas'' between two successive
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of ray (optics), rays that come to a focus (optics), focus ...

s is 2. * The celestial latitude (declination) of the Sun during a planet's astronomical cross-quarter day points equals the tilt of the planet's axis divided by .

*
List of mathematical constantsA mathematical constant A mathematical constant is a key number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. ...
*
Square root of 3 The square root of 3 is the positive real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican ... , * Square root of 5 The square root of 5 is the positive real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican ...
, * Gelfond–Schneider constant, *
Silver ratio In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
,

# References

* . * * Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI. * . * . * . * .